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VI.2 Efecto Josephson
VI Superconductividad
VI.1 Nociones generales y modelos teóricos
VI.2 Efecto Josephson y circuitos superconductores
con uniones Josephson
VI.3 Aplicaciones de la Superconductividad
VI.4 Prácticas
Juan José Mazo Torres
Agustín Camón
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VI.2 Efecto Josephson
Bibliografía
– Charles Kittel. Introduction to Solid State Physics. Wiley.
– N. W. Aschcroft and N. D. Mermin. Solid State Physics. Saunders
College (1976).
– Michael Tinkham. Introduction to Superconductivity (2nd
Edition). Mc Graw Hill (1996).
– Terry P. Orlando and Kevin A. Delin. Foundations of Applied
Superconductivity. Addison-Wesley (1990).
–A. Barone and G. Paternó. Physics and Applications of the
Josephson Effect. Wiley (1982).
–K. K.Likharev, Dynamics of Josephson Junctions and Circuits.
Gordon and Breach (1986).
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VI.2 Efecto Josephson
Outline
Basics
– Josephson effects
– Superconducting tunnel junction
– Modeling Josephson arrays
Localised excitations in Josephson arrays
–
–
–
–
Vortices in 2D arrays
Charge solitons in arrays of ultra-small junctions
Kinks in parallel arrays
Discrete breathers in JJA’s
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VI.2 Efecto Josephson
Josephson effect
(B.D. Josephson. Possible new effect in superconductive tunneling. Phys. Lett. 1, 251 (1962))
Josephson effect in weakly coupled macroscopic quantum systems:
•
9
Weakly coupled superconductors
•
•
•
•
9
A. Barone. Weakly coupled macroscopic quantum systems: likeness with difference, in
Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, edited I.O.
Kulik (Kluwer Academic, 2000)
A. Barone and G. Paternò. Physics and applications of the Josephson effect, Jonh Wiley, 1982.
K.K. Likharev, Dynamics of Josephson junctions and circuits, Gordon and Breach Science, 1984.
T.P. Orlandoand K.A. Delin, Foundations of applied superconductivity, Addison Wesley, 1991.
M. Tinkham, Introduction to superconductivity, Mc Graw Hill, 1996.
Weak coupling of superfluids
•
O. Avenel and E. Varoquax, Josephson effect and quantum phase slippage in superfluids, Phys.
Rev. Lett. 60, 416-419 (1988).
9
Weakly coupled Bose-Einstein condensates
•
A. J. Legget, Bose-Einstein condensation in the alkali gases: some fundamental concepts. Rev.
Mod. Phys. 73, 307 (2001)
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VI.2 Efecto Josephson
Superconducting (tunnel) Josephson junction
Superconductor
(Nb, Al,...)
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Insulating
~10Å (Al2Ox,...)
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
5 mm
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VI.2 Efecto Josephson
Superconducting (tunnel) Josephson junction
Superconductor
(Nb, Al,...)
Insulating
~10Å (Al2Ox,...)
Tunneling of Copper pairs between two superconductors?
Before 1962: no possible. Tunneling of single electrons has low probability, so simultaneous
tunneling of both electrons in a pair is statistically insignificant.
After 1962: In 1962 Josephson(1) showed that the Cooper pair is like a single particle, is the
macroscopic wave function that tunnels, so this is an observable process. Experimentally
confirmed by P. Anderson and J. Rowell(2) in 1963.
(1) B. D. Josephson. Possible new effects in superconductive tunneling. Physics Letters, 1, 251-253 (1962).
(2) P. W. Anderson and J. M. Rowell. Probable observation of the Josephson superconducting tunneling
effect. Phys. Rev. Lett. 10, 230 (1963)
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
Different approaches
1.- Microscopic theory (Josephson)
2.- Quantum Macroscopic Model
(Orlando)
3.- Quantum Macroscopic Model
(Feymann)
4.-Ginzburg-Landau theory
(for microbridges, Tinkham)
Different types of junctions
1.- SIS Tunnel junctions
2.- Weak links: SMS junction, microbridges, point contacts…
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
Superconducting (tunnel) Josephson junction
Superconductor
(Nb, Al,...)
Ψ1 = n1 e iθ1
Ψ2 = n2 e iθ 2
Insulating
~10Å (Al2Ox,...)
• Equations for the Josephson effect (1)
I S = I C sin ϕ
h dϕ
V =
2 e dt
r
2e 2 r
ϕ = θ 1 − θ 2 − ∫ A( r , t ) ⋅ d l
h 1
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• DC Josephson effect (V=0, ϕ=cte, I=cte≠0)
• AC Josephson effect (V≠0, ϕ=2eVt/h+ϕ0,
ac current with frequency ω=2eV/h)
Φ0 = h / 2e ; flux quantum
frequency:
483.6GHz / mV
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VI.2 Efecto Josephson
• Equations for the Josephson effect (2)
The microscopic theory gives more complex relations that in the case
of constant voltage V give:
I ( t , V , T ) = I J 1 ( V , T ) sin ϕ + I J 2 ( V , T ) cos ϕ + I qp ( V , T )
ϕ (t) = ϕ 0 + ω f t
with
ω
f
=
2e
V
h
I = I J 1 sin ϕ
Si
V = 0 then
I qp = I J 2 = 0
Si
V ≠ 0 then
I = I J 1 sin ϕ + V σ ( 1 + ε cos ϕ )
y
It includes an additional phase-dependent dissipative term reflecting an interference term
between the pair and the quasi-particle currents. This term averages out to zero in many
situations and in general is rather subtle to detect. Thus, is usually neglected.
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VI.2 Efecto Josephson
• Equations for the Josephson effect (3)
• Biased by a constant current I<Ic, then
ϕ=sin-1(I/Ic) and V=0
I S = I C sin ϕ
h dϕ
V =
2e dt
• Biased by an ac current Idc+Iacsin(ω0t), then V(t)=Vdc+ harmonics and we
can expect synchronization effects between the Josephson characteristic
frequency ωJ=2eVdc/h and the biasing frequency
[Vdc=(h/2e)ωJ=(h/2e)(n/m)ω0] which allows for a very precise determination
of the e/h ratio.
• This synchronization effect is also well observed when V=Vdc+Vaccosω0t,
then ϕ=ϕdc+ωJt+(2eVac/hω0)sin(ω0t) and
I s = I c ∑ (−1) n J n (2eVac / hω0 ) sin(ϕ dc + ω J t − nω0t )
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VI.2 Efecto Josephson
SIS junction
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SNS junction
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
• Josephson energy
U J = − EJ cosϕ
EJ =
hI c
2e
We need Josephson energy to exceed thermal energy: EJ»kBT (Ic»2ekBT/h)
• Temperature dependence of IC
Ambegaokar-Baratoff equation for the
temperature dependence of Ic in a
tunnel junction [PRL 10, 486 (1963);
erratum 11, 104 (1963)]
I c Rn = (π ∆ 2e ) tanh (∆ 2 k B T )
T=0 ⇒ IcRn=π∆(0)/2e (∆(0)=1.764kBTc)
After E. P. Balsamo at al., PRB 10, 1881 (1974)
T→Tc ⇒ (2.34πkB/e)(Tc-T)
(in SNS junctions Ic goes to zero at Tc and rises exponentially with decreasing temperature)
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
• Current-voltage characteristics: The RCSJ model.
I = I J + IC + I R
I
I J = I c sin ϕ ; I C = C
; with
V
dV
and I R =
R
dt
Equivalent circuit for the JJ
I
I c sin ϕ
+
1
I = C V& + V + I c sin ϕ
R
V
C
R
Φ ⎞
⎛
⎜V = 0 ϕ& ⎟
2π ⎠
⎝
i = ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ
1
1
where Γ = = 2 =
Q βc
Φ0
2π I c CR 2
I
; i=
and ω p =
Ic
2π I c
Φ 0C
Γ is the junction effective damping,
Q is called quality factor (also Q=ωpRC)
βc is called Stewart –McCumber
parameter
W.C. Stewart, Appl. Phys. Lett. 12, 277 (1968); D.E. McCumber, J. Appl. Phys. 39, 3113 (1968).
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VI.2 Efecto Josephson
Equivalent circuit for the JJ
+
I
I c sin ϕ
Non-linear pendulum
τ app
V
l
C
R
1
C V& + V + I c sin ϕ = I
R
ϕ
mg
ml 2ϕ&& = τ app − D ϕ& − mgl sin ϕ
ϕ&& + Γϕ& + sinϕ = I
Particle in a washboard potential
U (ϕ ) = − E J cos ϕ − (h I 2 e )ϕ
(mass and viscous damping)
m = (h 2 e ) C
2
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η = (h 2 e )2 (1 R )
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VI.2 Efecto Josephson
[Overdamped case: V=R(I2-Ic2)1/2]
3
2
Γ=5
IC
i=I/Ic
1
IV curves: RCSJ model vs
experiments
0
-IC
-1
-2
-3
-3
-2
-1
0
1
2
4
3
2
1
I/Ic 0
-1
-2
-3
-4
3
V/IcR=Γ<dϕ/dt>
[Underdamped case]
i=I/Ic
1
Γ=0.2
IC
Iret
0
V/Vg
-Iret
-1
-1,5
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5
g
-IC
-1,0
-0,5
0,0
0,5
1,0
V/IcR=Γ<dϕ /dt>
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(V
1,5
(I
ret
= 2∆ e )
≈ 4Γ π )
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VI.2 Efecto Josephson
Nonlinear subgap resistance
4
3
2
1
I/Ic 0
-1
-2
-3
-4
C V& +
1
V + I c sin ϕ = I
R (V )
⎧ Rn if V > 2 ∆ (T ) / e
R (V ) = ⎨
⎩ R sg (T ) if V < 2 ∆ (T ) / e
Rsg (T ) ≈ Rn e ∆ / k BT
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5
V/Vg
This quasiparticle tunneling current for kBT<<∆ and V<Vg=2∆/e is well approximated by
2G n − ∆ kT ⎛ 2 ∆ ⎞
⎛ eV ⎞ ⎛ eV ⎞
=
e
⎟
⎟K0 ⎜
⎟ ( eV + ∆ ) sinh ⎜
⎜
e
⎝ 2 kT ⎠ ⎝ 2 kT ⎠
⎝ eV + 2 ∆ ⎠
12
I qp
For some investigations and applications, it is convenient to shunt the junction by a small
resistance, then the Γ>>1 and the overdamped limit of the RCSJ model is appropriate. This
is also the case of SNS junctions.
In other cases, especially when dealing with small junctions, in order to describe the
behaviour of the system, it is essential to consider also the impedance of the external circuit.
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VI.2 Efecto Josephson
Thermal fluctuations
+
1
~
C V& + V + I c sin ϕ + I ( t ) = I
R
⎛ ~
⎜ I ( t ) = 0 and
⎝
I
I c sin ϕ
V
C
R
~
I(t)
2k T
⎞
I~ ( t ) I~ ( t ′ ) = B δ ( t ′ − t ) ⎟
R
⎠
-
~
i = ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ + i
ω(I)
∆U(I)
⎛ ~
⎜⎜ i (τ ) = 0 and
⎝
⎞
kT
~ ~
i (τ ) i (τ ′) = 2 Γ
δ (τ ′ − τ ) ⎟⎟
EJ
⎠
Overdamped junctions
V. Ambegaokar and B.I. Halperin,, Phys. Rev. Lett. 22, 1364 (1969)
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VI.2 Efecto Josephson
Underdamped junctions
Reduction of Ic by premature switching and
increasing of Iret by premature retrapping.
⎧
⎪
⎪
Γ(I ) =
⎨
⎪
⎪⎩
⎛ 18η
⎜
⎝ 5π
⎞⎛ ∆ U
⎟ ⎜⎜
⎠⎝ k B T
⎞
⎛ ∆U ⎞
⎟⎟ exp ⎜⎜ −
⎟⎟
⎠
⎝ k BT ⎠
IC
ctu
Flu
⎛ ∆U ⎞
⎛ ωa ⎞
⎟⎟
⎜
⎟ exp ⎜⎜ −
⎝ 2π ⎠
⎝ k BT ⎠
⎛ ωa2
⎜
⎜ 2πη
⎝
low damping
I
moderate damping
⎞
⎛ ∆U ⎞
⎟ exp ⎜ −
⎜ k T ⎟⎟
⎟
B
⎝
⎠
⎠
on
ati
s
Iret
V
high damping
In a typical experiment: IV curve where I is ramped from zero at a given rate:
ω a = ω p [1 − ( I / I c ) 2 ]1 / 4
∆ U ≈ 2 E J (1 − I / I c ) 3 / 2
{ [
Fluctuation effects cause a major reduction in Ic as
soonn an kBT is as large as 5 percent of EJ
] }
I c (T ) = I c 1 − (k B T / 2 E J ) ln (ω p ∆ t / 2π )
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VI.2 Efecto Josephson
AC fields
1
I = C V& + V + I c sin ϕ = I dc + I ac cos( ω t )
R
ℵ(ϕ ) = ϕ&& + Γϕ& + sinϕ = idc + iac cos(ω 'τ )
•Biased by an ac current Idc+Iacsin(ωt), then V(t)=Vdc+ harmonics and we can
expect synchronization effects between
the Josephson characteristic frequency
ωJ=2eVdc/h and the biasing frequency.
They are the so-called Shapiro steps.
Overdamped case: no hysteresis
Underdamped case: hysteresis
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VI.2 Efecto Josephson
Long Josephson-junction
A long JJ is a junction which has one
dimension (say x) long with respect to the
Josephson penetration depth λJ. Then the phase
difference is also a function of the spatial
coordinate ϕ(x,t).
The electrodynamics of the junction is described by a nonlinear partial differential equation that,
neglecting dissipative effects, can be written as
ϕ xx − ϕ tt = sin ϕ
This is the sine-Gordon equation, which has coherent
localised particle-like solutions or soliton solutions. ϕ(x) can
be interpreted as the phase difference or the normalized
magnetic flux. Then a soliton in the junction corresponds to
solution for which the junction goes from 0 to 2π or the flux
goes from 0 to Φ0, that is a fluxon of magnetic field.
When losses and bias are includes the dynamics of the fluxon is described by the perturbed
sine-Gordon equation
ϕ xx − ϕ tt − sin ϕ = αϕ t − βϕ xxt − γ
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VI.2 Efecto Josephson
Quantum JJ
Superconductor
(Nb, Al,...)
Ψ1 = n1eiθ
1
Ψ2 = n2 eiθ
Insulating
~10Å (Al2Ox,...)
2
• Junction Hamiltonian
Josephson energy
Charging energy
Q2
UC =
2C
e2
= EC
2C
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UJ = −
Q2
H = − E J cos ϕ +
2C
hIc
cosϕ
2e
EJ
(Q ≈ d ϕ / dt )
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VI.2 Efecto Josephson
To study the quantum dynamics of the junction we will follow the canonical quantization
rules and treat ϕ and Q as operators which satisfy the usual communtation rule,
[ϕ , Q ] = i 2e
Then, in phase representation,
∂
Q
∂
ϕ
= n→ i
⇒
H
=
−
E
cos
−
4
E
J
C
∂ϕ 2
2e
∂ϕ 2
2
2
By analogy with the problem
of a particle in a periodic
potential the solutions of this
Hamiltonian will take the
form of Bloch functions.
The ratio EJ/4EC measures the relative importance of the charging energy of the pairs. When
EJ is dominant the quantum fluctuations in the phase are small. When EJ ∼EC, the kinetic term
induce a delocalization of the phase. In the other limit EJ <<EC tunneling is weak, and large charge
transfers are energetically prohibitive.
In order to observe quantum behavior we need to have the thermal energy kBT<<EJ,EC
and large tunnel resistances R>h/4e2=6.45kΩ
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VI.2 Efecto Josephson
Evidence of “Secondary” Quantum effects
9
Macroscopic quantum tunneling of the phase and energy level
quantization [J. Clarke et al., Science 239, 992 (1988)]
9
Single Cooper-Pair tunneling [L.J. Geerligs et al., PRL 65, 378 (1990)]
9
Demonstration of Heisenberg’s uncertainty principle in a superconductor
[W.J. Elion et al., Nature 371, 594 (1994)]
9
Dissipative phase transition in a single Josephson junction
[Penttilä et al., PRL 82, 1004 (1999)]
Operation of quantum states with single JJ and simple JJ circuits
9
Quantum superposition of distinct macroscopic states in a rf SQUID [J.R. Friedman et al.,
Nature 406, 43 (2000)] and a superconducting loop with 3 junctions [C.H. Van der Waal et al.,
Science 290, 773 (2000)]
9
Manipulation od the quantum state of superconducting tunnel junction Circuit
[D. Vion et al, Science 296, 886 (2002)]
9
Generation and observation of coherent temporal oscillations between the macroscopic
quantum states of a Josephson tunnel junction [Y. Yu et al, Science 296, 889 (2002)]
9
Josephson vortex qubit [A. Kemp et al, PSS B, 233, 472 (2002)]
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VI.2 Efecto Josephson
Modeling Josephson arrays
Φ 0 dϕ
2π dt
I J = I c sin ϕ
V=
Ψ2 = Ψ2 e iθ 2
Ψ1 = Ψ1 e iθ1
2e 2 r r
ϕ = θ 1 − θ 2 − ∫1 A dl
h
E = E J (1 − cos ϕ ), with E J =
SQUIDs
I
Parallel
Series
x
x
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I I I I I I
x
I
Ladder
Φ0
Ic
2π
I
I
I
2D
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VI.2 Efecto Josephson
• Kirchoff’s conservation law (for the current and for the voltage)
• Fluxoid quantization: for any loop l in the array (with at least one junction) the sum
of all the phase differences around the loop is
∑ ϕ = 2π ( n − f )
j∈l
j
l
l
The integer nl is the vorticity of the loop and results from the multivaluedness of the
superconducting phase variable (θ)
The term fl accounts for the total (external plus induced) magnetic flux through the loop
measured in terms of Φ0 (fl =Φl/ Φ0). In general, to compute the total induced flux in a cell one
must take into account a full inductance matrix. However, in many cases we can work with the
simplest approximation of taking into account only self-inductances (L) of each cell. The
parameter λ= Φ0 /2πIcL measures the importance of induced fluxes
1 I loop
ext
ind
ext
We will leave out the nl terms and write ∑ ϕ j = − 2π ( f l + f l ) = − 2π f l −
λ Ic
j∈l
Josephson circuits can be divided into two general categories. Circuits of the first type have
λ>>1 so the induced flux in the loop is not important (these circuits are typically made of
aluminium). Circuits of the second type have λ<<1 and induced flux causded by circulating
currents is important (these circuits are typically made of niobium).
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VI.2 Efecto Josephson
Series Arrays
N
V ( t ) = ∑ϕ&k = F ( I L ( t ))
k =1
I
LOAD
ϕ&&k+ Γϕ&k + sinϕ k + I L ( t ) = I
[P. Hadley at el., PRB 38, 8712 (1988)]
Search for junction dynamic synchronization for coherent radiation and
definition of the voltage standard.
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VI.2 Efecto Josephson
rf-SQUID (the superconducting loop interrupted by a JJ)
ϕ = −2π
L
Φ
λ=
1
βL
Ic
=
Φ
Φ0
⎛
Φ⎞
⎟⎟
Φ = Φ ext − LI c sin⎜⎜ 2π
Φ
0 ⎠
⎝
⎛
Φ⎞
⎟⎟
I s = I c sin⎜⎜ 2π
⎝ Φ0 ⎠
2
⎛
⎛
Φ ⎞ ⎞⎟ (Φ − Φ ext )
⎜
⎜
⎟
U = EJ ⎜1 − cos⎜ 2π
⎟⎟ +
2L
⎝ Φ0 ⎠ ⎠
⎝
Φ0
2πI c L
Important for basic reasearch (Schrödinger cat problems,
Caldeira-Legget model for quantun dissipation,…) and
detection of magnetic fields. Operated at a radio frequency
(typically between 20 and 30 MHz). An external circuit
biased by an rf field and inductively coupled to the SQUID.
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VI.2 Efecto Josephson
dc SQUID ( a) neglecting inductive effects)
Fluxoid quantization
If inductive effects can be neglected, fluxoid quantization imposes a
constraint to the equations and then reduces the number of
independent variables of the system,
1
2
Iext
(ϕ1 − ϕ 2 ) = − 2π
Φ0
Current conservation
Φ ext = −2πf 0
I ext = I1 + I 2 = I c sin ϕ1 + I c sin ϕ2 = 2I c cos(πf 0 ) sin(ϕ1 + πf 0 )
Behaves like a single junction with a critical current modulated by the external field !!
Then the IV curve for the system, in the overdamped regimen (resistively shunted
junctions) is given by:
V=(R/2){I2-[2Iccos(πΦext/Φ0)]2}1/2
Measuring the critical current or the voltage change at a given bias current we can
measure the external field.
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VI.2 Efecto Josephson
dc SQUID ( b) with inductive effects)
Fluxoid quantization
(ϕ1 − ϕ 2 ) = −2πΦ tot / Φ 0 =
1
= −2π (Φ ext + Φ ind ) / Φ 0
L
Current conservation
I1 = I c sin ϕ1 = − I m + I ext / 2
I 2 = I c sin ϕ1 = I m + I ext / 2
2
Φind = LI m = L( I 2 − I1 ) / 2 = LI c (sin ϕ2 − sin ϕ1 ) / 2
Iext
If Iext is very small, then ϕ1 and ϕ2 are almost
zero and we can write
Φext=Φ+LIcsin(πΦ/Φ0)
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VI.2 Efecto Josephson
dc SQUID (a different approach)
Current conservation
Iext
i1 = ϕ&&1 + Γϕ&1 + sinϕ1 = i mesh + iext
ϕ1
I
m
B app
ϕ
i2 = ϕ&&2 + Γϕ& 2 + sinϕ 2 = − i mesh + iext
2
Fluxoid quantization
(ϕ 1 − ϕ 2 ) = − 2π
Φ0
L
i
mesh
(Bapp A + LI mesh )
= − λ (ϕ 1 − ϕ 2 + 2π f 0 )
f 0 = Bapp A Φ 0
λ = Φ 0 2πLIc
ϕ&&1 + Γ ϕ& 1 + sin ϕ 1 = − λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext
ϕ&&2 + Γ ϕ& 2 + sin ϕ 2 = λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext
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VI.2 Efecto Josephson
2-junctions in parallel
τ app
Iext
ϕ1
2-coupled pendula
Im
ϕ
B app
λ
2
ϕ1
ϕ
2
L
ϕ&&1 + Γ ϕ& 1 + sin ϕ 1 = − λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext
ϕ&&2 + Γ ϕ& 2 + sin ϕ 2 = λ (ϕ 1 − ϕ 2 + 2π f0 ) + iext
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VI.2 Efecto Josephson
JJ parallel array
Iext
Im
Im
Im
Bapp
Bapp
ϕj −1
Bapp
Im
ϕj
Im
Bapp
ϕj +1 B
app
Im
Bapp
• Set of junctions connected
in parallel by superconducting
wires.
L
τapp
ϕ
j −1
ϕ
• All the junctions have the
same dc voltage
λ
j
ϕ
j+1
ϕ&& j + Γ ϕ& j + sin ϕ j = λ (ϕ j +1 − 2ϕ j + ϕ j −1 ) + iext
(Forced and damped Frenkel-Kontorova model or discrete sine-Gordon equation)
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VI.2 Efecto Josephson
JJ ladder array
• Vertical junctions are connected
in parallel by horizontal junctions.
Nonlinear pendula coupled by
nonconvex springs.
• Not all the vertical junctions have
the same dc voltage
ξ j = ϕ jv + ϕ jt − ϕ jv+1 − ϕ jb + 2π f 0 = − 2π f jind
(we have considered no external field)
Φ0
=λ
2π I cy L
λ
Φ0
λx =
=
2π I cx L h
λy =
Física de Bajas Temperaturas
ℵ(ϕ j ) = −
t
λ
ξj
h
ℵ(ϕ jv ) = λ (ξ j −1 − ξ j ) + iext
ℵ(ϕ j ) =
b
h=
λ
h
ξj
I cx C x R y
=
=
I cy C y Rx
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VI.2 Efecto Josephson
2D array
λ
ext
i
ℵ(ϕ ijx ) = (ξ ij − ξ ij −1 ) + x
h
h
ℵ(ϕ ijy ) = λ (ξ i −1 j − ξ ij ) + i yext
ξ ij = − 2π f ijind = ϕ ijy + ϕ ijx+1 − ϕ iy+1 j − ϕ ijx + 2π f0
( i , j + 1)
i xext
ϕ ij
y
( i, j )
( i, j )
ϕ ijx
Φ0
=λ
2π I cy L
λ
Φ0
λx =
=
2π I cx L h
λy =
( i + 1, j )
i yext
This is a model for a 2D array when only self-inductances are taken into account.
Sometimes, depending on the problem to be studied, inductances are not needed at
all; in other cases however a full inductance matrix is necessary.
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
• In general, in a square 2D array we have to solve equations for the dynamics of 2N2-2N (for FBC)
gauge-invariant phase differences, ϕij. However, when induced fields can be neglected (λ>>1 limit),
flux quantization condictions imposes (N-1)2 constrains on these variables. Then, it is more
convenient to express the system equation in terms of the phase in each island, θi, which are N2-1
independent variables.
ϕ ij = θ i − θ j −
r
2π j r
A( r ) ⋅ d l
∫
Φ0 i
the dynamical equations result after applying current conservation.
• In the EJ>> EC limit the total Josephson
energy is all the relevant energy:
H J = − ∑ E J cos(θ i − θ j − Aij )
< ij >
where Aij depend now only
on the external magnetic
field
• In the opposite limit of ultrasmall junctions (EC>> EJ), the charging energy is the more
important contribution:
HQ =
1
∑ij Q i C ij−1Q j
2
Qj stands for charge in island j. Cij for the
capacitance matrix of the circuit.
• The most interesting case is that of intermediate values where both terms should be
considered and
H =
1
∑ij ( Q i + qi )C ij−1 ( Q j + q j ) − <∑ij > E J cos( θ i − θ j − Aij )
2
qj are possible offset charges or charges induce by external sources (charge frustration). Charge and
phase variables are canonically conjugate variables.
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VI.2 Efecto Josephson
Localised excitations in Josephson arrays
The concept of coherent structures or coherent excitations has important
consequences when applied to condensed matter systems. Spatially or temporally
coherent structures appear in many nonlinear extended systems. Such structures
usually can be characterized by marked particle-like properties. In the past few years,
these notions have become fundamental for understanding many problems and their
implications extend over different fields of the physics of continuous and discrete
systems. JJ arrays are a very well example of an experimental system where such type
of structures appear examples that we are going to present are: Vortices, kinks and
breathers.
– Vortices in 2D arrays
– Charges and charge solitons in arrays of ultra-small
junctions
– Kinks in parallel arrays
– Discrete breathers in JJA’s
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VI.2 Efecto Josephson
Vortices in 2D arrays
[The two-dimensinal physics of Josephson junction arrays. R.S. Newrock, C.J. Lobb,
U. Geigenmüller and M. Octavio. Solid State Physics, Vol 54, pp 263-512 (2000)]
In classical 2D arrays of superconducting islands coupled by “large” JJs the relevant
energy is the sum of the Josephson energies of the junctions
HJ = −
∑ E J cos( θ i − θ j )
(no magnetic fields)
< ij >
• This sytem is a realization of the 2D XY model, a system which shows the KTB phase transition
where vortex-antivortex pairs unbind.
• The model supports....
¾ “spin waves”: small amplitude and energy wave-like variations of the phase over the array
¾ vortices (and antivortices): localised excitations defined by the sum of the phase
differences along a path containing the vortex whis is equal to 2π (-2π for antivortices).
[Σ(θi−θj)=2πn] Vortices and antivortices are topological excitations, they behave large
charges.
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VI.2 Efecto Josephson
Single vortex properties at zero temperature
• Energy of the ground state E=-2N2EJ
• Energy of an isolated vortex in a large square
array of size L=Na→∞ E=πEJln(L/a)-2N2EJ
• An external current can move the vortex
F=(Φ0/2π)(Itot/Na)=(Φ0/2π) (I/a)
• Energy barrier for a single vortex: in a square
lattice EPN=0.2EJ ; for triangular lattices,
EPN=0.043EJ
• 2D PN potential. The potential along x direction
UPN(x)=-(EPN/2) cos(2πx/a) [Vortex depinning
current Icv=(π/Φ0) EPN=0.1Ic]
• Effective equations for the motion of a vortex:
external force (external current), viscous force
(power dissipated in the resitive channels),
kinetic energy (stored in the capacitors),
potential energy (Josephson energy)
Φ 0 d ( 2π x a ) Φ 0C d 2 ( 2π x a )
I = I cv sin( 2π x a ) +
+
4π R
dt
4π
dt 2
Física de Bajas Temperaturas
[From Newrock et al Sol. Stat. Phys. 54, 263 (2000)]
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VI.2 Efecto Josephson
Array properties at non zero temperatures
• There are not free vortices at low enough
temperatures [the energy to add a vortex is
EV=πEJln(L/a) (valid for large L)]
• At T≠0 the thermal generation of bound
pairs of vortices is energetically favored
[energy of a v-av pair Ep=2πEJln(r/a)]
• Free vortices appear above KTB
temperature, transition temperature TKT ∼
πEJ/2kB
These results are affected by the inclusion
of finite size effects, external magnetic
fields, self-induced fields, disorder,... but in
any of the cases the concept of vortices is
essential to understand the physics of the
array.
Física de Bajas Temperaturas
[From Mooij and Schön in “Single Charge
Tunneling” Ed. By H. Grabert and M.H.
Devoet, NATO ASI Series B: Physics Vol.
294,Plenum Press NY (1992)].
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VI.2 Efecto Josephson
Charges in 2D arrays of ultrasmall junctions
[H. J. Mooij and G. Schön, Single charges in 2-dimensional junction arrays. In “Single Charge
Tunneling”, p. 275, ed. By H. Grabert and M.H. Devoret, Plenum Press NY (1992)]
In arrays of ultrasmall tunnel junctions the
more relevant energy is the charging energy
HQ
1
= ∑ Q i C ij−1Q j
2 ij
• Qi is the charge in the island. For metalic “normal state” islands this charge is an integer multiple
of the electronic charge e. In the superconducting state, at low temperatures and voltages below
the gap voltage, Cooper pairs are dominant and the charge appears in multiple of 2e.
• In any of the cases at zero temperature no free charges are present in the array. The system is
insulating. In arrays where mutual capacitance are dominant (C>>C0) the charges interact
logarithmically over sufficient long distances. Then free charges are created by thermal activation
through a KTB phase transition where pair of charges e, -e unbind and the array is resistive
(Tcn=EC/4πkB, Tcs=EC/πkB, EC=e2/2C).
• This transition is strongly affected by dissipation [measured by the coefficient αT=h/(4e2RT)],
offset charges, ...
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VI.2 Efecto Josephson
Charge-vortex duality
• In small capacitance superconducting junctions we have to include both the charging
and the Josephson coupling. If we ignore quasiparticles the hamiltonian of the system
is
1
H =
Q i C ij−1Q j −
E J cos( θ i − θ j )
2 ij
< ij >
∑
∑
[we have ignored also charge or phase frustration (offset or induced charges and magnetic fields)]
If charging energy can be ignored EC=0, the vortices undergo a KTB transition where vortex
dipoles unbind. This transition separates a superconducting low temperature phase from a
resistive high temperature phase. If the Josephson coupling is weak (EJ=0) the charges show a
KTB transition where the dipoles, formed by a Cooper pair and a missing pair, unbind. The
transition separates an insulating from a conducting phase.
At finite EJ and EC both charge and vortex excitations have to be considered simultaneously.
The charging energy provides a kinetic energy for the vortices, and the Josephson coupling
allows the tunneling of cooper pairs and provides dynamics for the charges (If EC<<EJ or
EJ<<EC perturbative approaches can be used).
However, to fully understand the physics of the system we have to consider also the influence
of the quasiparticle tunneling. We have a quantum dissiptive system.
• We have also seen that vortices can be described as particles moving in a substrate potential
with a mass given by the charging energy. For small junctions we can expect strong quantum
mechanical efects. At finite temperatures vortex motion can be thermally excited. At low
temperatures it is possible to observe quantum mechanical tunneling of vortices.
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VI.2 Efecto Josephson
Charge Solitons in 1D arrays
[P. Delsing, One-dimensional arrays of small tunnel junctions. In “Single Charge Tunneling”
Ed. By H. Grabert and M.H. Devoret, p. 249. Plenum Press NY (1992) and D.B. Haviland and P.
Delsing, “Cooper-pair charge solitons: The electrodynamics of localized charge in a
superconductor”, Phys. Rev. B, 54, R6857 (1996)]
• Weak Josephson coupling EJ<<EC and large normal resistance R>h/4e2
• If a single electron (or single pair) is added to or substracted from an
intermediate island the resulting localized state -the single charge plus the
polarization cloud- is called charge soliton (or anti-soliton). It correspond to a
localized voltage profile. Some of their properties have been studied with the use
of a sine-Gordon equation.
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VI.2 Efecto Josephson
Kinks in parallel arrays
[L.M. Floría and J.J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Adv. Phys. 45, 505 (1996). O.Braun
and Y. Kivshar, Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Rep. 306, 1 (1998). A.V. Ustinov et al, Fluxon
dynamics in one-dimensional Josephson-junction arrays, 47, 8357 (1993). S. Watanabe et al, Dynamics of circular arrays
of Josephson junctions and the discrete sine-Gordon equation, Physica D 97, 429 (1996). F. Falo et al, Fluxon ratchet
potentials in superconducting circuits, Appl. Phys. A 75, 263 (2002).]
Iext
Im
Im
Im
Bapp
Bapp
ϕj −1
Bapp
ϕj
Im
Im
Bapp
ϕj +1 B
app
Im
Bapp
L
Discrete sine-Gordon equation
or Frenkel-Kontorova model
ϕ&& j + Γ ϕ& j + sin ϕ j = λ (ϕ j +1 − 2ϕ j + ϕ j −1 ) + iext
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VI.2 Efecto Josephson
The system supports different types of excitations:
¾ “linear waves”: small amplitude and energy wave-like variations of the
phase over the array
When the phases are small, the solution to the
linearized form of the equations (for zero damping
and current) are: ϕj(t) α exp i(ωkt-kj) with
ωk2 = 1 + 4λ sin2 ( k / 2)
( −π < k < π )
They are linear waves with a frequency spectrum
characterized by a finite band with gap ωmin=ω(k=0)=1
and maximun frequency ωmax=ωπ=(1+4λ)1/2
¾ when the phases are not small the linear approximation is not valid and
the dynamics is pretty rich supporting new type of localized excitations: kinks
and breathers
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VI.2 Efecto Josephson
A kink(anti-kink) -discrete solitons, elementary discommensurations, ...corresponds to a solution where the phases go from 0 to 2π(−2π) along the
array. Since (ϕj+1-ϕj)=-2πΦj/Φ0 ; then (ϕN-ϕ1)=2π=-2πΦtot/Φ0 ; one kink
corresponds to one fluxon in the array.
9 They are the only excitations that can exist in the static state.
9 The existence of kinks do not depend crucially on the discreteness of the system (in
many aspects they are similar to the solitons found in the continuous version of the
equation, the sine-Gordon equation).
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VI.2 Efecto Josephson
9 However the existence of a discrete lattice breaks the invariance under continuous
translations of the solitons of the continuous model. The kink is invariant under discrete
translations. Due to discreteness the kink is pinned to the lattice. Then there exist a minimun
energy barrier the kink needs to overcome to move through the lattice. This energy is the socalled Peierls-Nabarro barrier EPN.
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VI.2 Efecto Josephson
We can go further and define not only a PN barrier
but a PN potential for the kink.
To compute the kink potential profile we let the kink
configuration to relax (overdamped dynamics) from the
saddle-point configuration to the minimun energy
configuration. During the relaxation we work out the
energy and center of mass of the configuration
obtaining the potential profile E(XCM)
ϕ& j = − sin ϕ j + λ (ϕ j +1 − 2ϕ j + ϕ j −1 )
•
•
•
λ
E
= ∑ (1 − cos ϕ j ) + (ϕ j +1 − ϕ j ) 2
j
2
EJ
X CM = C ± ∑ ϕ j
j
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VI.2 Efecto Josephson
Then, in some cases is possible to identify the kink motion with the motion of a single
particle over a (almost sinusoidal) periodic potential defined by:
E PN
(1 − cos X )
VPN ( X ) =
2
•
In the presence of external bias, an effective equation of motion for the kink is given by
E PN
&
&
&
mX + mΓX +
sin X = i
2
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VI.2 Efecto Josephson
The picture of the kink or fluxon as a single particle is useful in arrays which are larger
than the fluxon width and are not driven or driven by small currents (far from the
whirling mode).
When the kinks move it radiates energy in form of small amplitude waves. This radiation
is very strong for the case of underdamped arrays. There phonons are easily excited by
the kink in its wake and resonances between the kink velocity and these waves appear.
A particularly interesting configuration is a
ring of JJ connected in parallel. There, once
the array is superconducting magnetic field
gets trapped as an integer number of
fluxons.
[After S. Watanabe et al, Phys. Rev. Lett. 74,
174 (1995)].
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VI.2 Efecto Josephson
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VI.2 Efecto Josephson
We have considered arrays where all the junctions are identical and all the cells
have the same size. However, it is possible to design different arrays which are
adequated for studying new physical problems.
One example is the use of Josephson parallel arrays to study the dynamics of kinks
subjected to substrate ratchet potentials [F. Falo et al, Fluxon ratchet potentials in
superconducting circuits, Appl. Phys. A 75, 263 (2002)].
A ratchet potential is a periodic potential
without inversion symmetry: V(x) = V(-x),
then it’s easier to move a perticle in one
direction than in the other.
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VI.2 Efecto Josephson
Examples of the PN potential for a fluxon trapped in a ring
E/EJ
4,5
4,5
(a)
4,0
4,0
3,5
3,5
3,0
3,0
13
14
15
16
17
(c)
3,5
13
3,0
2,5
2,5
2,0
2,0
14
15
16
14
15
16
17
13
17
(d)
3,5
3,0
13
(b)
14
15
16
17
XCM
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VI.2 Efecto Josephson
Fluxon depinning currents for the non-ratchet and the ratchet parallel arrays.
1,00
Ratchet array
0,2
0,75
∆Idep/Ic
Idep/Ic
0,1
0,50
Ratchet array
0,0
0,0
0,5
λ
1,0
1,5
Regular array
0,25
0,00
0,0
Física de Bajas Temperaturas
0,5
1,0
1,5
λ
2,0
2,5
3,0
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